Chapter 4.7, Problem 47E

In quantum mechanics, the study of the Schrodinger equation for the case of a harmonic oscillator leads to a consideration of Hermite’s equation,

$y^{\prime \prime }-2t{y}^{\prime }+\lambda y=0$,

where $\lambda$ is a parameter. Use the reduction of order formula to obtain an integral representation of a second linearly independent solution to Hermite’s equation for the given value of $\lambda$ and corresponding solution $f\left(t\right)$.

a. $\lambda =4$, $f\left(t\right)=1-2t^2$

b. $\lambda =6$, $f\left(t\right)=3t-2t^3$